Let a(x), b(x), p(x) be formal power series in the indeterminate x over [Formula: see text] (i.e. elements of the ring [Formula: see text] of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group [Formula: see text] in [Formula: see text]. By a covariant embedding of the linear functional equation [Formula: see text] (for the unknown series [Formula: see text]) with respect to [Formula: see text]. In this paper we solve the system ((Co1), (Co2)) (of so-called cocycle equations) completely, describe when and how the boundary conditions (B1) and (B2) can be satisfied, and present a large class of equations (L) together with iteration groups [Formula: see text] for which there exist covariant embeddings of (L) with respect to [Formula: see text].

We study local analytic solutions f of the generalized Dhombres functional equationholomorphic in some open neighborhood of 0, depending on f , and f (0) = w 0 . After deriving necessary conditions on ϕ for the existence of nonconstant solutions f with f (0) = w 0 we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w 0 is not a root of 1. If |w 0 | = 1 or if w 0 is a Siegel number we show that all formal solutions yield local analytic ones. For w 0 with 0 < |w 0 | < 1 we give representations of these solutions involving infinite products.

We study local analytic solutions f of the generalized Dhombres equation f ðx f ðxÞÞ ¼ 'ð f ðxÞÞ with f ð0Þ ¼ 0 in the complex domain. We give an existence result, describe the structure of the set of all local analytic solutions and solve the converse problem, i.e., we characterize those local analytic functions which are solutions of a generalized Dhombres equation. Connections of generalized Dhombres equations with linear functional equations and generalized Böttcher equations are used. Furthermore, we establish relations of generalized Dhombres equations with Briot-Bouquet differential equations and with iteration groups. Finally, as an application of Böttcher functions, we describe the connections between two generalized Dhombres equations and the representations of their solutions as infinite products. (2000): Primary 30D05, 34M25, 39B12, 39B32; Secondary 30B10. Mathematics Subject Classification

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